This would be tricky, I admit, but hear me out.
I've found the low levels of the clockworks to be a little too easy for me, but I'm still stuck running them to accumulate crowns and materials.
I'd love if I could optionally turn up the difficulty (more than just doing a Danger Room) for the opportunity to get a few extra crowns or materials.
For example, perhaps if a group of players can clear a floor using only charged attacks, they earn a bonus of 10% crowns or get a small pot of level-dependent materials. Or perhaps the challenge is "Don't take damage", or "Don't use shields", or have to use a single weapon and not switch for that level. Maybe the challenge is presented to the players by that floor of the dungeon and they can choose to take it or ignore it.
My only concern is that a mixed group of players might have difficulty if one player opts to break the challenge to save his or her skin and angers his or her allies. It also presents a dilemma for players who may join in the middle of a level and get rewarded for a challenge they did not complete (though this is a minor concern).
It also may confuse newbie players who are not yet skilled enough to survive with all their options available, let alone when under a self-imposed handicap.
Alternately, you could offer handicaps as an option for when the player chooses to start the run, and players choosing to join in mid-run will have to specifically request to join the handicapped group.
Alternately, breaking handicap results in immediate disqualification of the player who breaks it, but not his or her allies which prevents concerns of trolling by ruining handicap games in progress. This also prevents one player from breaking a handicap purposely but sticking around to pull the other players through (effectively cheating the handicap).
This idea is inspired by the Conducts of Nethack, which I consider to be at least part of the inspiration for Spiral Knights' design.